Defining Chaos

Hi guys,

Sorry for the late post. Batterman’s paper was quite dense for me since it was full of concepts I hadn’t encountered before and so took me a while to digest them.

In this paper Batterman criticises Stone and Ford separately. In this post I am going to mention some points which came to my mind during reading his critique of Stone (I may post another note for Batterman’s critique of Ford in the next couple of days) and comparing it with lecture slides.

1- Stone introduces three criteria for deterministic system: (a) there exists an algorithm which maps a state of system at any given time to a state at any other time (not probabilistic), (b) a given state is always followed by the same history of state transitions, (c) any state of system can be described with arbitrarily small nonzero error.

Batterman says condition (c) which is known as well-posedness is equivalent to the notion of continuous dependence which says: a solution depends continuously on the initial data if convergence of initial data entails convergence of solutions. It is not transparent to me that these two are equivalent. To make it clear we first need to find out what Stone means by the vague term “describe” in condition (c). We may want to say that by this term he means 'a description of the states of system which uses algorithm and some other states'. So we may reconstruct condition (c) as: 'any state of system can be described, with arbitrarily small nonzero error, using algorithm and some other state of system'.

Let’s compare this reconstruction with condition (d) which Stone introduces for defining the notion of predictability: "any state of the system can be generated from the algorithm with arbitrarily small nonzero error from any other state of the system”. These two sentences seem very similar except for their second quantifiers. 

However, it seems, condition (d) and reconstruction of condition (c) are far from precise mathematical definition of continuous dependence and non-sensitive dependence which, in the paper, are respectively taken to be equivalent to (c) and (d). The mathematical definition of continuous dependence is in the slides, and the mathematical definition of non-sensitive dependence is the following:

∀δ, ∃ε, x₀, ∀x₁,t such that

dist(x₀ , x₁) < ε → dist(φ(x₀, t), φ(x₁, t)) < δ

One can see that condition (d) is far from this definition, at least in terms of its details: in condition (d) there is no mention of time variable. I don’t know why Batterman doesn’t criticise Stone for this.

2- Stone claims  that determinism is a necessary condition for predictability. Batterman provides an example of an indeterministic system which is predictable to falsify Stone’s claim: "one can make decent predictions about quantum systems”. My question is: Can one make decent non-probabilistic predictions about quantum systems? If not, then I doubt this does count as a counterexample for Stone’s claim as I think what Stone means by prediction is non-probabilistic prediction.

3- Stone says "systems which do not admit closed form solutions have this property that the error present in the specification of the initial state can be amplified exponentially". The problem is Stone doesn’t provide any argument for this claim and I couldn’t find any criticism about this in Batterman's paper.

4- I pondered for a while on the difference between discontinuous dependence and sensitive dependence and finally reached the conclusion that having (ε, δ)-definition of the mathematical notion of limit in mind, one can say that the difference between discontinuous and sensitive dependence is analogous to the difference between (1) a situation in which a discontinuous function like f(x) doesn’t have any limit when x approaches a number like c and (2) another situation in which the limit of a function like g(x)  goes to infinity as x approaches c.

5- In the slides it was said that exponential unstable systems may have continuous dependence but Batterman says they may have non-sensitive dependence. I was wondering which one of these claims is true and how one can prove either of them by using the mathematical definition of exponential instability: dist (φ(x₀,t), φ(x₁,t)) ≥ (dist (x₁,x₀))^λt

P.S: I realised that in the paper Batterman does not assert that Stone's condition (d) is equivalent to non-sensitive dependence. So I was wrong in saying that these two are taken to be equivalent throughout the paper (However he explicitly claims condition (c) and continuous dependence are equivalent). He also doesn't mention anything about the relation between this condition and continuous dependence. He only claims requirement (d) can be satisfied by exponential unstable systems. So I was also wrong in claiming that Batterman says exponential unstable systems may have non-sensitive dependence.

Dainton, Teller, and the construction of neo-Newtonian spacetime

I told the class last week that I would look again at Dainton’s discussion of the order in which neo-Newtonian spacetime is constructed and the charge of emptiness against that construct.  I forgot to mention it in class this week, so here are my thoughts.

I still think that it is not literally the order in which the elements or components of neo-Newtonian spacetime are constructed that is really at issue here.  Paul Teller argues that the inertial frames and trajectories of neo-Newtonian spacetime are distinguished only by their lack of inertial effects (effects like the tension on the cord between Newton’s globes, the tendency of the water in Newton’s bucket to rise up the sides of the bucket, and the feelings of pressure, weight, and perhaps queasiness that we feel in a rapidly rising airliner).  Since it is (supposedly) only on the basis of lacking such effects that we pick out the inertial trajectories, Teller claims that we are not really explaining the lack of such effects (or their presence in other cases) by pointing out that those in which the effects do not occur are inertial.  The suggestion seems to be that we are in effect saying nothing more than ‘there are no inertial effects associated with these trajectories because there are no inertial effects associated with them’.   

Perhaps this can be understood as a matter of order:  We first observe that certain trajectories or frames are not associated with inertial effects, and we then label these the inertial frames/trajectories.  But what really matters in Teller’s argument is not the order but the basis, i.e., the reason for distinguishing the trajectories in question.  In principle, we might have called them inertial first, and then justified this later in terms of the lack of inertial effects.

In any case, I think Teller and Dainton have both overlooked something important here.  It is not as if just any old trajectory counts as inertial just because of an observed lack of inertial effects.  In fact, the inertial trajectories all have a special relation to one another:  They all exhibit uniform relative velocity to each other.  I think this suggests that when we say that these trajectories are not associated with inertial effects because they are inertial trajectories, we are not just blurting out an empty tautology, but identifying something special about a particular class of trajectories, something that is plausibly due to the particular structure of the spacetime they inhabit and their relation to that structure.

Announcement:  I am about to post a corrected version of the differential equations "handout".  The arguments in Equation (2) were reversed.  Thanks to Somayeh for catching that. 

On the Notion of Cause 2

Hi again! 

So rather than posting another long summation I’m just going to point out a 5 interesting points about Russell’s meaning of cause and effect.

-         -          The idea of a time-constant, which Russell considers essential, is in fact dependent on multiple assumptions as to the meaning of both cause and effect. Does Russell go far enough in his explanation to be able to justify this?

-          The sentence “What is essentially the same statement of the law of causation” isn’t allowed, why is Russell allowed to get away with that??? He builds the second half of his critique off this assumption (making some selective interpretations, something he then attacks Bergson for).

-          Does the fact that an ‘event’ is only likely to occur once have any relevance to the possibility of a universal law of causality? For me it’s a separate point.

-          “I deny is that science assumes the existence of invariable uniformities of sequence of this kind, or that it aims at discovering them”. This encapsulates Russell’s view on the matter, and his whole argument is based on this belief.

-          “The principle "same cause, same effect," which philosophers imagine to be vital to science, is therefore utterly otiose.” I don’t think that Russell’s points really prove this. Surely he is only pointing out problems with this assumption but I don’t he think he goes as far as invalidating it.

Hope these statements are suitably provocative!!! 

On The Notion Of Cause

Hi everybody! Sorry for the late post!

On the notion of cause

In this paper Russell discusses the notion of cause as used in science and philosophy. He states that his three aims as the following,

1.       To discuss the flaws in the use of the term ‘cause’ in a scientific context
2.       Suggest a more appropriate alternative to the ‘law of causality’
3.       Explore some of the problems in theology and determinism that result from the improper use of ‘cause’

This first post will discuss the first of these points as the others depend on the assumption of the first. 

The cause of an event is often considered to be a crucial part of scientific inquiry, with the likes of Ward suggesting that causes are indeed the very ‘business of science’. In this piece Russell contends that not only are ‘causes’ not the subject of advanced sciences but that in reality there is no such thing. He first questions the appropriateness of the word cause by using the then available definitions of the word, taken from Baldwin’s dictionary. These are as follows,

(1)    CAUSALITY - The necessary connection of events in the time-series

(2)    CAUSE (notion of) - Whatever may be included in the thought or perception of a process as taking place in consequence of another process

(3)    CAUSE AND EFFECT - Cause and effect are correlative terms denoting any two distinguishable things, phases, or aspects of reality, which are so related to each other, that whenever the first ceases to exist, the second comes into existence immediately after, and whenever the second comes into existence, the first has ceased to exist immediately before

Russell discusses the appropriateness of each of these in turn (and at length) so let us begin with the first.

Causality:

From the statement it is clear that the meaning of causality is dependent on the meaning of necessary. Hence to fully understand the implications of cause it is first required that we determine the meaning of necessary. Necessary is defined as follows,

NECESSARY - That is necessary which not only is true, but would be true under all circumstances. Some-thing more than brute compulsion is, therefore, involved in the conception; there is a general law under which the thing, takes place

Russell points out several problems with this statement. To understand Russell’s critique it is required that we first understand his ideas of a proposition and a propositional functions.

Proposition – A statement which is either true or false, no other considerations can apply

Propositional Function – A statement containing a variable, which is called the argument. Assigning a value to the variable creates a proposition (the statement must then be either true or false).

[I’m curious as to whether these were well established concepts or are they of Russell’s own creation. This would seem to have a significant impact on the validity of his argument, given that it relies on Baldwin making an informed inclusion of the terms in his definition.]

With these definitions in mind let us re-examine the definition of necessary. According to Baldwin for something to be necessary it must be ‘true in all circumstances’. This implies that the subject must therefore be a propositional function rather than a proposition. The example given is that ‘if x is a man, then x is mortal’. As all men are mortal any value for x which satisfies the first parameter must therefore satisfy the second. Russell suggests an alternative definition for necessary on this basis,

NECESSARY – “is a predicate of a propositional function, meaning that it is true for all possible values of its argument or arguments.”

However Baldwin’s definition states that for something to be necessary it must also be true. This is impossible to rectify with Russell’s ideas of propositions and propositional functions (which are mutually exclusive) as the definition requires that a statement be both at the same time (impossible because whilst a statement can become a proposition from a propositional function it could not under Russell’s definition be both at the same time).

[My personal thoughts on this are that Russell is being ‘unnecessarily’ literal with Baldwin’s definition. Whilst I accept the point that there is a conflict, it would appear more obvious (and convenient!) to dismiss Baldwin’s double use of true as mere emphasis. Dismissing the either, allows the definitions of causality and necessary to fit.]

Russell resolves this conflict by requiring the argument of the function to be identified within the statement. He therefore arrives at the following definition which satisfies the implied meaning of causality,

NECESSARY - A proposition is necessary with respect to a given constituent if it remains true when that constituent is altered in any way compatible with the proposition remaining significant.

Armed with this definition Russell gives what he interprets to be the universal law of causality (whilst making 100% sure he is not tied to its authenticity!)

Cause (Notion of):

Russell doesn't really spend much time discussion the 2^nd definition as it does not really concern the meaning of cause which is of interest to him (so I won’t either).

Cause and Effect:

Definition 3 appears to be the closet to what is traditionally meant by causality in my opinion (or at least when it is used in reference to science). The problem with the statement according to Russell is the implied temporal contiguity. Treating the current and prior states as distinct entities introduces the requirement for causal relations between the two.

[The implications of this get a bit wordy but I’ll do my best to simplify what I think Russell is trying to say.]

The definition seems to describe cause and effect as having an associated finite time (not really instantaneous). Hence within the duration of the cause there can be said to be earlier and later parts. If this is taken to be true then only the later stages of the cause, those immediately preceding the effect, can be truly relevant to the effect. The earlier parts of the cause are not contiguous to the effect and so could therefore be altered without changing the effect. Really you’re supposed to think that the effect is only dependent on a certain final part of the cause.

This is clearly a bit ridiculous, and does not fit what cause is supposed to be (at least in my opinion). Russell agrees and says it is hardly acceptable that the effect should just spring from a cause at some point. Cause and effect cannot therefore be contiguous in time. Russell then introduces the time-interval to solve this.

I reckon this is enough for a first blog as it covers Russell’s critique of Baldwin’s definitions. I shall put a second post up tomorrow which will discuss the proposed solution to the 3^rd definition.

Maudlin and the spatiotemporal shifts

Hello all, 

I've limited the scope of this post to Maudlin's conception of Newtonian substantivalism and how it copes with the static and kinetic spatiotemporal shift arguments. I had difficulty grasping the content of sct. 4, and as such I think it would be more productive to focus on sct. 3, which is relatively straight forward but yields some interesting questions and responses. I'm not entirely sure of the best way to structure this, so I will briefly summarise the text, highlighting any questions or concerns that occurred to me. 

Maudlin offers a view on the classical debate of which holds that the key issue under consideration, the fact that Newtonian absolute space has a "metaphysically distasteful" character of ontologically distinct but empirically identical circumstances, is brought about by a fundamental misunderstanding and subsequent misuse of the arguments. He claims that the weight of Leibniz' spatiotemporal shift arguments is down to a fallacious equivocation, and that upon investigation into their structures, one finds that they are essentially different, and therefore cannot be legitimately appealed to. 

The arguments in brief: In a substantivalist universe, where spatiotemporal properties of objects are defined absolutely in relation with absolute space, and defined locally in relation with one another, a static shift in time and/or space would essentially be unnoticeable, since every observation made before and after the shift would be identical. Similarly, a kinematic shift, in which the universe in its entirety would undergo motion with repeat to absolute space, would also be unnoticeable for the same reason. If the relations between bodies would not change, the observations made within such a shifted universe would still hold as they had before, and therefore the two universe pre-and-post-shift would be indiscernible from one another. Following the Principle of the Identity of Indescernibles, the two universes would be identical. "In short, both the static and kinematic shifts (...) would result in ontologically distinct but observationally indistinguishable states of affairs."

Maudlin goes on to expose the failure of the argument, which rests in an “illusions engendered by imprecision concerning the notion of observationally indistinguishable states of affairs.” The key here is that what is observationally indistinguishable in the case of the kinematic shift is not necessarily so in the case of the static shift. Essentially, within Newtonian space-time, observation is possible if there is a change in spatiotemporal property with respect to anything other than absolute space, so if two shifted universe states differ only with respect to absolute space, the difference will not be observable, and therefore the states will be observationally indistinguishable. What this seems to come down to is that in the case of a kinematic shift, we can say that we part of the actual world, but there are certain aspects of it that are unknowable.

Similarly, in the case of the static shift, there will be no way to observe certain spatiotemporal properties that would help us identify which state of affairs we are part of. Maudlin then poses the following question: if the entire universe were to shift statically in space or time, what could we say about the actual state of the universe? In the kinematic case, he says, physical questions about the actual nature of the universe can be asked but not necessarily answered, whereas in the static case they cannot even be asked in the first place without indexically picking out a spatiotemporal location, and identifying the occupant of that location is a physical question that can be answered.

Maudlin provides an example for this which amounts to the claim that it knowledge of the spatiotemporal locations of things is presupposed in referring to them. Were the universe shifted statically 15 billion years into the future, he says, it is possible that someone just like him would be sitting in the same place writing the same words, however the indexicals used in his writing would ensure that the referents of his utterances would not be the same. In that sense, it is an error on the part of Leibniz to assume that a static shift would be undetectable; not because it could be observed, but because the observer would no longer be around to observe it.

I agree with this to some degree in the case of a temporal shift, but I am not convinced the same applies in the spatial shift. It is possible that the entire universe from my point of view could shift spatially billions of lightyears to the left, but if the relative distances and positions between the occupants of the universe do not change, that is, were their positions to change only with respect to “absolute” space, then the spatial location contained in the meaning of my utterances of “here” or “there” would also have to change, and there would be no observation possible that would make me aware of that. In other words, it is true that “here” means two different things pre-and-post-shift, but it seems dubious that my knowledge of that should track the shift, as it were. In Maudlin’s words, “We can (…) formulate meaningful counterfactuals about worlds where everything would be displaced from its actual location, but we can also be assured that they are counterfactuals.” Where does this assurance come from? It seems to me that taking spatiotemporal properties as counterfactuals presupposes that the actual state can be discerned through observation, which is precisely what remains to be shown.

Where Maudlin loses me is in his conception of what a static shift would be like, so to close this post I would like to examine the ways in which this is conceivable. The point of the static shift argument is to show that in absolute space, a spatiotemporal change of position of the entirety of the universe, that is, everything that exists within it, with respect to absolute space, would be unnoticeable, and that therefore a view of space-time which holds on to absolute space but admit the possibility of an infinity of possible universes. But if we ignore PII (which, as Maudlin notes, is no longer taken seriously), it seems that this is not a problematic consequence.

Questions and answers about neo-Newtonian spacetime.

Hi all,

Emma emailed me some questions about neo-Newtonian spacetime, and I thought it would be helpful to post my answers here.   Please feel free to add remarks or follow-up questions in the comments.

--MWP

----------------

1. When you were going through the list of things that might be permitted under neo-Newtonian spacetime, I understood all of the answers except for the last one - if your spaceship is moving 100m/s relative to my spaceship why is that permitted if objects do not have spatial relations through time? Is it because we can manipulate the hyperplanes so that my spaceship could also be moving 100m/s relative to your spaceship? If so, I'm still not sure how 'm/s' is a permissible unit of measurement? To me, in neo-Newtonian spacetime 'metres-per-second' are forbidden or nonsensical.

Answer:  It's not totally obvious, but we can reconstruct relative speed in neo-Newtonian spacetime.  We have simultanaeity, distance at a time, and elapsed time.  So two bodies (spaceships, say) have a distance at one time and another distance at another time.  Lets say they have distance d at time t and distance d' at time t'.  Then over the time interval from t to t' they have an average speed relative to each other, which is just the difference in distance divided by the difference in time:  (d' - d) / (t' - t).  So in neo-Newtonian spacetime we can reconstruct average relative speeds between bodies.
     Notice that we get this "average" without adding up a bunch of values and then dividing by the number of values.  We just take the ratio of the final differences.  In fact, instantaneous speeds are derived from the average speeds by looking at averages over tiny time intervals.  So we can also reconstruct instantaneous relative speeds in neo-Newtonian spacetime.  We just don't have absolute speeds.  Measuring the distance between the spaceships at different times won't tell us which one is moving or which one is moving faster.

2. Is space still absolute in neo-newtonian spacetime? If so, why can't objects move absolutely? I'm just picking up on your line: "Neo-Newtonian space time does not distinguish between absolute velocities." I think I have misunderstood the sentence. Does it mean that an object can move absolutely, but since we have no unit of measurement for velocity in the neonewtonian spacetime, there can be no such thing as a 'difference' in velocity between one object's worldline and another's'?

Answer:  In neo-Newtonian spacetime, space is still absolute in the sense that, within each instant (each spatial hyperplane) there are different places.  There is still an intrinsic difference between being here and being there.  But objects cannot move absolutely because we have here no concept of being in the same place over time.  Motion is change of place over time, and that makes no sense here.  There is no rule (or even an unruly map) to identify a point in space now with a point in space at another time.  So the statement "Neo-Newtonian spacetime does not distinguish between absolute velocities" means that, in neo-Newtonian spacetime, bodies don't even have absolute velocities, and there is not even a distinction between rest and uniform motion.  Quite aside from the matter of stipulating a unit of measure for velocity, we can't even distinguish between some velocity and none at all.  But, as explained above, there is such a thing as a difference in velocity, or a relative velocity.  A body A has a definite relative velocity to any body B, but neither A nor B has a definite velocity on its own.

 3. (continuation from 2) I think my problem is just trying to conceptualise the scenario: if absolute acceleration is permitted, but absolute velocity is not, what is the line moving through as it goes between hyperplanes, if not spatial distance? If I walk from here to the door in a neo-Newtonian spacetime, and am accelerating, my world line will curve upwards through various hyperplanes, yet at the same time, my worldline has not travelled any 'distance'. So here's my attempt to understand it: my worldline is moving through the hyperplanes as if through a series of pictures - so it's more like a dot-to-dot through fixed moments, than a representation of any smooth transition through time resulting in a spatial relation. Is that it?

Answer:  No, a smooth transition is the right picture.  Between any two hyperplanes there are (in the neo-Newtonian model) infinitely many more hyperplanes, forming a smooth continuum.  As we pass from one hyperplane to another, we are always in some hyperplane.  And I think you should think of a curved worldline as representing motion through spatial distance.  It's just that you can't pick out any particular part of the curve as being nearly at rest or as being in very fast motion.  There is no specific distance between temporally separate events, but there are relations between temporally separate events.  The permissible alignments of spatial hyperplanes is constrained by the fact that all inertial worldlines must remain straight.  So you might say there is sort of a loose relation of distance over time, even though it is not fully pinned down.  In that sense, bodies do move through space, just not at any particular speed.
     On the other hand, it might be better to think of bodies as moving through spacetime rather than through space.  The curve traced by a body moves from sheet to sheet in a particular way, even though the distances between points on different sheets are undefined.  It is not moving left-to-right or east-to-west, but forward in spacetime, in a manner that traces either a straight or a curved path.
     I hope that helps, but I think that, with this question, we are dealing with cognitive discomfort more than questions of fact.  You may have to just relinquish certain habits of thought.

Mitch on Newton

Newton’s Scholium on time, space, place and motion

These are a few general questions that came up when I was reading Newton and Rynasiewicz and Newton’s arguments for absolute motion, as I think they could be reconstructed:

Newton’s aim is to show that true motion must be absolute by showing that it cannot be relative to other bodies. Rynasiewicz, Newton and the lecture slides describe it as to show that motion cannot be determined, or analysed, by only looking at objects and how they move relative to each other. How is that different from the claim that true motion is absolute motion? Is this epistemic aspect weaker, or does this aspect only follow from Newton’s premises?

There are some formulations in both texts which to me seem to indicate that Newton argues for the identity of true and absolute motion. For example, Rynasiewicz states that “the difficulty of (…) so doing constitutes for Newton a strong argument for the existence of absolute space”. The step from absolute space to absolute motion is not far, Newton even defines absolute motion as a change of place in absolute space, early in the Scholium (IV). Another one is that Newton, according to Rynasiewicz, wants to justify his distinction of true and relative motion” – meaning true motion is not relative, thus absolute.

It seems strange to me that Newton defines true motion as the change of place through absolute space. That suggests that he defines true motion to be absolute motion and then uses this definition for circular arguments. That true motion exists is agreed on by all, but absolute space and motion should be his conclusions, rather than dogmatically fixed in the beginning. Throughout the text, both Newton’s formulations and the structure of his arguments look as if this was the case. Some of his premises appear to just formulate negations of Descartes’ views.

Furthermore, according to Rynasiewicz, Newton treats “absolute motion” and “true motion” as synonyms (in 5.3). This does not make sense to me at all. Either that is intended, but then the arguments are circular, or Newton is not clear enough.

I have tried to reconstruct the four arguments from properties, cause and effect, and will also shortly comment on their appeal to me.

Properties 1:

I found this argument the hardest to understand. My first intuitive reconstruction was this:

1.       Property: If two things are at true rest, then they are also rest with respect to each other

2.       If true rest is taken to be relative (in the way Descartes does), then two things can be at true rest, but not with respect to each other (they are then only at rest relative to their direct neighbours but not to each other).

3.       So, true rest cannot be taken to be relative.

4.       If true rest cannot be relative, it must be absolute.

5.       So, true rest must be absolute.

The first premise is convincing, if it is interpreted as the following answer to why one should believe it to be true: It is our strongest intuition corresponding to the way we speak about motion and rest, that if two things are at true rest, they are also rest with respect to each other. This is just what we mean when we say “rest” and “motion”. To state the premise just as a fact is problematic, because it presupposes absolute rest. To defend Descartes, one could then simply reject the premise on grounds of Descartes’ notion of true rest.

But this reconstruction does not seem to really reflect Newton’s intention.

1.       Property: if two things are at true rest, then they are also rest with respect to each other.

2.       This property implies that in order to find that something is truly at rest one has to find that it is at rest relative to something at true rest, which cannot be done by analysing anything around us.

3.       Thus, one cannot analyse something around us in order to find true motion.

Unfortunately, this is the best I could do. This cannot be right, the second premise is clearly wrong: the property does not imply that, because one cannot find something by finding its necessary condition.

Properties 2:

1.       Property: If a part of an object moves, then another part of the same object moves with it.

2.       Two bodies where one surrounds the other one are like two parts of one body.

3.       So, if the surrounding body moves, the surrounded body moves with it.

4.       If true motion is motion relative to its contiguous bodies, then the surrounded body does not move.

5.       So, the surrounded body moves and does not move, which is impossible.

6.       Thus, true motion cannot be motion relative to its contiguous bodies.

It is natural to agree on the property, since a body as a whole either moves or does not move. I think the crucial premise is premise 1, the question whether that analogy really holds.

At first I reconstructed this argument in a different way, which does not seem to be what Newton aims at, but still looks like an interesting challenge of Descartes’ view similar to the above:

1.       If true motion is motion relative to its contiguous bodies, then a body, contiguous with two objects, one of which is moving and the other of which is not moving, moves and does not move at the same time.

2.       A thing cannot move and not move at the same time.

3.       So, true motion is not motion relative to its contiguous objects.

Is that convincing? Could Descartes still claim the predicate “to move” to be a one place predicate?

Argument from causes:

1.       Force impressed on a body is necessary and sufficient for its true motion (or the alteration of it).

2.       If true motion is motion relative to its contiguous bodies, then force impressed on a body is neither necessary nor sufficient for its true motion (the alteration).

3.       Thus, true motion cannot be motion relative to its contiguous bodies.

The question here is whether 1 is true. This is again a premise that Newton just seems to take for granted, but that seems intuitively right, just like the properties in argument 1 and 2.

Argument from effect:

The argument consists of a counterexample on Descartes’ universal claim that all true motion is relative to contiguous bodies.

1.       The centrifugal force is an effect and therefore a necessary and sufficient condition of rotation, which is true motion.

2.       The rotating bucket experiment shows that if true motion is motion relative to its contiguous bodies, then there can be a truly moving (rotating) object, that does not show any centrifugal force (water at the beginning) and that there can be a truly resting object with clear signs of centrifugal force impressed on it. The effect is neither necessary nor sufficient here for motion.

3.       So, true motion cannot be motion relative to its contiguous bodies.